High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature

Although the Bock–Aitkin likelihood-based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations, a serious limitation of the method is its reliance on fixed-point Gauss-Hermite (G-H) quadrature in the solution of the likelihood equations and likelihood-ratio tests. When the number of latent dimensions is large, computational considerations require that the number of quadrature points per dimension be few. But with large numbers of items, the dispersion of the likelihood, given the response pattern, becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space. In this paper, we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data. In particular, we show that adaptive G-H quadrature, combined with mean and covariance adjustments at each iteration of an EM algorithm, produces an accurate fast-converging solution with as few as two points per dimension. Evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of upto eight dimensions. Unlike an earlier application of adaptive Gibbs sampling to this problem by Meng and Schilling, the simulations also confirm the validity of the present method in calculating likelihood-ratio chi-square statistics for determining the number of factors required in the model. Finally, we apply the method to a sample of real data from a test of teacher qualifications.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43596/1/11336_2003_Article_1141.pd

A Parameterization for Individual Human Growth Curves

Using data from the Fels growth study, we show that individual curves for growth in recumbent length from one year to maturity can be represented in good approximation by the sum of two logistic components. The first component describes growth occurring throughout the prepubertal period and continuing in some degree until maturity; the second describes the adolescent growth spurt. The components are functions of six parameters, five of which are estimated by non-linear least squares, and the sixth is the mature stature taken directly from the data. A reliability analysis of the parameter estimates for the Fels samples shows that most of the individual differences in growth pattern, within sex, can be attributed to three, or at most four, of the six parameters. Distributions of estimates of these four parameters are presented and discussed in relation to sex differences

A Parameterization for Individual Human Growth Curves

Using data from the Fels growth study, we show that individual curves for growth in recumbent length from one year to maturity can be represented in good approximation by the sum of two logistic components. The first component describes growth occurring throughout the prepubertal period and continuing in some degree until maturity; the second describes the adolescent growth spurt. The components are functions of six parameters, five of which are estimated by non-linear least squares, and the sixth is the mature stature taken directly from the data. A reliability analysis of the parameter estimates for the Fels samples shows that most of the individual differences in growth pattern, within sex, can be attributed to three, or at most four, of the six parameters. Distributions of estimates of these four parameters are presented and discussed in relation to sex differences